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Number Systems

Number Systems. Common Number Systems. Quantities/Counting (1 of 3). Quantities/Counting (2 of 3). Quantities/Counting (3 of 3). Conversion Among Bases. The possibilities:. Decimal. Octal. Binary. Hexadecimal. Decimal to Decimal (just for fun). Decimal. Octal. Binary. Hexadecimal.

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Number Systems

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  1. Number Systems

  2. Common Number Systems

  3. Quantities/Counting (1 of 3)

  4. Quantities/Counting (2 of 3)

  5. Quantities/Counting (3 of 3)

  6. Conversion Among Bases • The possibilities: Decimal Octal Binary Hexadecimal

  7. Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal

  8. Weight 12510 => 5 x 100 = 5 2 x 101 = 20 1 x 102 = 100 125 Base

  9. Binary to Decimal Decimal Octal Binary Hexadecimal

  10. Binary to Decimal • Technique • Multiply each bit by 2n, where n is the “weight” of the bit • The weight is the position of the bit, starting from 0 on the right • Add the results

  11. Example Bit “0” 1010112 => 1 x 20 = 1 1 x 21 = 2 0 x 22 = 0 1 x 23 = 8 0 x 24 = 0 1 x 25 = 32 4310

  12. Decimal to Binary Decimal Octal Binary Hexadecimal

  13. Decimal to Binary • Technique • Divide by two, keep track of the remainder • First remainder is bit 0 (LSB, least-significant bit) • Second remainder is bit 1 • Etc.

  14. 2 125 62 1 2 31 0 2 15 1 2 3 1 2 7 1 2 0 1 2 1 1 Example 12510 = ?2 12510 = 11111012

  15. Common Powers (1 of 2) • Base 10

  16. Common Powers (2 of 2) • Base 2 • What is the value of “k”, “M”, and “G”? • In computing, particularly w.r.t. memory, the base-2 interpretation generally applies

  17. / 230 = Example In the lab…1. Double click on My Computer2. Right click on C:3. Click on Properties

  18. Review – multiplying powers • For common bases, add powers ab ac = ab+c 26 210 = 216 = 65,536 or… 26 210 = 64  210 = 64k

  19. Binary Addition (1 of 2) • Two 1-bit values “two” pp. 36-38

  20. Binary Addition (2 of 2) • Two n-bit values • Add individual bits • Propagate carries • E.g., 1 1 10101 21+ 11001 + 25 101110 46

  21. Multiplication (2 of 3) • Binary, two 1-bit values

  22. Multiplication (3 of 3) • Binary, two n-bit values • As with decimal values • E.g., 1110 x 1011 1110 1110 0000 111010011010

  23. Fractions • Binary to decimal 10.1011 => 1 x 2-4 = 0.0625 1 x 2-3 = 0.125 0 x 2-2 = 0.0 1 x 2-1 = 0.5 0 x 20 = 0.0 1 x 21 = 2.0 2.6875

  24. Fractions • Decimal to binary .14579x 20.29158x 20.58316x 21.16632x 20.33264x 20.66528x 21.33056 etc. 3.14579 11.001001...

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